last product will be the length of the arc; for, when the radius is 1, balf the circumference is 3,14159265, &c. 3,14159265 and therefore, =,01745329, or ,0174533, nearly 2 180 degrees, which is the length of an arc of i degree. Hence CDX ADBX b= the length of the arc, ADB. EXAMPLE. (17) What is the length of the arc, ABD, which is 29,5 de grees, and radius 93 PROBLEM IX. RULE. Multiply the radius by half the arc of the sector, found by the last problem, and the product will be the area, as in the whole circle. AB Fig 9. EXAMPLE. (18) What is the area of a sector, whose radius, CA, is 55, and the length of the arc, AB, 59 ? PROBLEM X. To find the area of the segment of a circle, ADB, whose centre is E. (See Fig. 8.) RULE. Find the area of the triangle ABC, by Prob. III. and of the sector, ADBC, by the last problem: their difference, or sum of these areas, will be that of the segment, according as it is less or greater than a semi-circle. Or, To six times the base AB (see Fig. 8.), add eight times the chord of half the arch AB, viz. DB; multiply the sum by the altitude DE, divide the product by 15, and the quotient will nearly give the area. A TABLE of the segments of circles, whose area is unity or 1, the diameter being divided by parallel chord-lines into 100 equal parts. V.S. Segment. V.S. Segment. ! V.S. Segment. | V.S. Segment. 26 28 ,0017 ,0048 ,0087 ,0134 ,0187 ,0245 ,0308 ,0375 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ,0446 ,052 ,0598 ,068 ,0764 ,0851 ,0941 ,1032 ,1127 ,1224 ,1323 ,1424 ,1526 ,1631 ,1738 ,1845 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 ,2066 ,2178 ,2292 ,2407 ,2523 ,2640 ,2759 ,2878 ,2998 ,3119 ,3241 ,3364 ,3486 ,3611 ,3735 ,3860 ,3986 ,4112 ,4238 ,4365 ,4491 ,4618 ,,4715 ,4873 ,5 17 18 19 20 51 ,5127 ,5255 ,6681 66 ,7002 67 :,7122 68 ,7241 69 ,756 70 ,7477 71 ,7593 72 ,7708 73 ,7822 74 ,7934 75 ,8045 76 ,8155 77 ,8262 78 ,8369 79 ,8474 80 ,8576 81 ,8677 82 ,8776 83 ,8873 84 ,8968 85 1,9059 86 ,9149 87 ,9236 88 ,932 89 ,9402 90 ,948 91 ,9554 92 ,0625 93 ,9692 94,9755 95 ,9813 96 ,9866 97 ,9923 98 ,9952 99 ,9983 100 11,0000 21 22 23 24 25 ,19.55 EXAMPLES. (19) Suppose the diameter, FG, of a circle to be 84 inches, and the height of the segment, ED, 30 inches, what will its area be? (20) What is the area of a segment whose arc is a quadrant, or contains 90 degrees, and diameter 18 feet? PROBLEM XI. To find the area of a segment of a sector, ABCD, or the front of an arch built with stones of equal length. RULE. Multiply half the sum of the bounding arches, AB and CD, by the distance, AC, and the product will give the area, AB+CD 2 (21) What is the area of the front of an arch built with stones 34 feet long, whose upper and lower bounding arches are in length 84 and 727 respectively? (22) What is the area contained between two concentric semi circles, whose diameters are 24 and 16? PROBLEM XII. To find the area of an ellipsis, or oval. RULE. Multiply continually together the two axes, and the number ,7854 (b), and the product of these three numbers will express the area. (23) What is the area of an ellipsis whose greatest diameter is 24, and the least diameter 18? OF ARTIFICERS' WORK. L. Glaziers' and Masons' Fiat Work is measured by the Foot Square. EXAMPLES. (1) What is the content of 12 panes of glass, each measuring 3 feet 10 inches long, and 2 feet 8 inches broad? What will the glazing come to at 8 d. per foot? (2) There is a bouse with 3 tier of windows, 4 in a tier; the height of the first tier is 6 feet 6 inches, the second 51 feet, and the third 4 feet; the breadth of each win. dow is 3 feet 9 inches. What will the glazing come to at 10d. per foot ? (3) What is the price of a marble slab, whose length is 6 feet, and the breadth 31 feet, at 8s. per foot : (4) A looking-glass is 16 inches by 9, and contains a foot of glass. What will the content of the plate be, that bas twice the length, and three times the breadth? II. PAINTING, PLASTERING, PAVING, &c. is measured by the yard square, which is 9 square feet. RULE. Divide the square feet by 9; and the quotient will be the number of square yards. |